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# Polynomial and Rational Functions

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```Polynomial and Rational
Functions
Lesson 2.3
Animated Cartoons
Note how
mathematics
are referenced
in the creation
of cartoons
Animated Cartoons
We need a way
to take a number
of points
and make
a smooth
curve
This lesson
studies
polynomials
Polynomials
General polynomial formula
P ( x ) пЂЅ a n x пЂ« a n пЂ­1 x
n
n пЂ­1
пЂ« ... пЂ« a1 x пЂ« a 0
вЂў a0, a1, вЂ¦ ,an are constant coefficients
вЂў n is the degree of the polynomial
вЂў Standard form is for descending powers of x
вЂў anxn is said to be the вЂњleading termвЂќ
Note that each term is a power function
Family of Polynomials
Constant polynomial functions
вЂў f(x) = a
Linear polynomial functions
вЂў f(x) = m x + b
Quadratic polynomial functions
вЂў f(x) = a x2 + b x + c
Family of Polynomials
Cubic polynomial functions
вЂў f(x) = a x3 + b x2 + c x + d
вЂў Degree 3 polynomial
Quartic polynomial functions
вЂў f(x) = a x4 + b x3 + c x2+ d x + e
вЂў Degree 4 polynomial
Properties of Polynomial Functions
If the degree is n then it will have at most
n вЂ“ 1 turning points
вЂў
вЂў
вЂў
End behavior
вЂў Even degree
вЂў Odd degree
or
or
Properties of Polynomial Functions
Even degree
вЂў Leading coefficient positive
вЂў Leading coefficient negative
Odd degree
вЂў Leading coefficient positive
вЂў Leading coefficient negative
Rational Function: Definition
Consider a function which is the quotient of
two polynomials
R ( x) пЂЅ
P ( x)
Q ( x)
Example:
r ( x) пЂЅ
2500 пЂ« 2 x
x
Both polynomials
Long Run Behavior
a n x пЂ« a n пЂ­1 x
n
Given
R ( x) пЂЅ
n пЂ­1
b m x пЂ« b m пЂ­1 x
m
m пЂ­1
пЂ« ... пЂ« a1 x пЂ« a 0
пЂ« ... пЂ« b1 x пЂ« b0
The long run (end) behavior is determined
by the quotient of the leading terms
вЂў Leading term dominates for
large values of x for polynomial
вЂў Leading terms dominate for
the quotient for extreme x
an x
n
bm x
m
Example
Given
3x пЂ« 8x
2
r ( x) пЂЅ
5x пЂ­ 2x пЂ«1
2
Graph on calculator
вЂў Set window for -100 < x < 100, -5 < y < 5
Example
Note the value for a large x
3x
2
5x
2
How does this relate to the leading terms?
Try This One
Consider r ( x ) пЂЅ
5x
2x пЂ« 6
2
Which terms dominate as x gets large
What happens to
5x
2x
2
as x gets large?
Note:
вЂў Degree of denominator > degree numerator
вЂў Previous example they were equal
When Numerator Has Larger Degree
Try
2x пЂ« 6
2
r ( x) пЂЅ
5x
As x gets large, r(x) also gets large
But it is asymptotic to the line
yпЂЅ
2
5
x
Summarize
Given a rational function with
leading terms
When m = n
вЂў Horizontal asymptote at
a
b
When m > n
вЂў Horizontal asymptote at 0
When n вЂ“ m = 1
вЂў Diagonal asymptote
yпЂЅ
a
b
x
an x
n
bm x
m
Vertical Asymptotes
A vertical asymptote happens when the
function R(x) is not defined P ( x )
вЂў This happens when the
denominator is zero
пЂЅ R ( x)
Q ( x)
Thus we look for the roots of the
2
denominator
x пЂ­9
r ( x) пЂЅ
x пЂ« 5x пЂ­ 6
2
Where does this happen for r(x)?
Vertical Asymptotes
Finding the roots of
the denominator
x пЂ« 5x пЂ­ 6 пЂЅ 0
2
( x пЂ« 6)( x пЂ­ 1) пЂЅ 0
x пЂЅ пЂ­ 6 or x пЂЅ 1
View the graph
to verify
x пЂ­9
2
r ( x) пЂЅ
x пЂ« 5x пЂ­ 6
2
Assignment
Lesson 2.3
Page 91
Exercises 3 вЂ“ 59 EOO
```
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